


A Mathematical Proof for Marriage

by Bitenomnom



Series: Mathematical Proof [1]
Category: Sherlock (TV)
Genre: Gen, M/M, Mathematics, Pining Sherlock, warning: heavy math usage
Language: English
Status: Completed
Published: 2012-09-04
Updated: 2012-09-04
Packaged: 2017-11-13 14:38:33
Rating: Not Rated
Warnings: Creator Chose Not To Use Archive Warnings
Chapters: 1
Words: 984
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/504561
Author URL: https://archiveofourown.org/users/Bitenomnom/pseuds/Bitenomnom
Summary: <blockquote class="userstuff">
              <p>The ranking system in Sherlock's mind was nice and well ordered, until one John Watson almost screwed it up.</p>
            </blockquote>





	A Mathematical Proof for Marriage

**Author's Note:**

> My attempt at motivating myself to both review my notes and practice writing regularly: a series of Sherlock drabbles that relate (however indirectly!) to the gradate math courses I'm taking. As a warning, they will not be beta'd, Brit-picked, or really in any way tested before posting. They will also probably be really short and (I hope) rather varied. My goal (roughly) is to do one for each day I have classes, i.e. four a week. I'm posting them to keep myself accountable, but would also be delighted to hear it if you have any feedback. (Or any Britpick, corrections, etc.)
> 
> Today: a little bit of light Johnlock (sort of). I shall include the math from which I got the idea, and sort of a translation. How did Sherlock get these math terms in his head? Dunno. Maybe he has a cursory knowledge of set theory. Or, like, a math book sitting open at the flat for a case.
> 
> (See end-notes for link to podfic and Chinese translation!)

Relation **_R_** on _E_ is called a _partial ordering_ if it is reflexive, antisymmetric, and transitive. In this case, ( _E_ , **_R_** ) is called a _partial ordered set_ (or, _poset_ ).

A poset ( _E_ , **_R_** ) is called a _well_ (or, _totally_ ) _ordered set_ or a _chain_ iff either _x ≤ y_ or _y ≤ x_ for any _x_ , _y_ ϵ _E._

A poset ( _P_ , **≤** ) is called an _upper semilattice_ iff _x_ V _y_ exists for any _x, y_ ϵ _P_ ; A poset ( _P_ , **≤** ) is called a _lower semilattice_ iff _x_ Λ _y_ exists for any _x, y_ ϵ _P_ ; A poset ( _P_ , **≤** ) is called a _lattice_ iff it is both an upper semilattice and a lower semilattice.

—Wang, Z., Yang, R., & Leung, K. (2010).  _Nonlinear integrals and their applications in data mining_. (Vol. 24, pp. 19-20). Singapore: World Scientific Publishing.

 

Translation: A relation on a set is called a _partial ordering_ if it is reflexive (for instance, a ≤ a), antisymmetric (for instance, if a ≤ b and b ≤ a, then a = b), and transitive (for instance, if a ≤ b and b ≤ c, then a ≤ c). The set ( _E_ , **_R_** ) of the set and the relation is called a _poset_. If any two points in the set can be compared, then the poset is _totally ordered_. If you can find an upper bound between any two points and a lower bound between any two points in a poset, the poset is known as a _lattice_.

 

***

Sherlock’s mind had always been a lattice—in the mathematical sense, of course, not the gardening sense.

He ordered things inside his mind. Constantly. Everything. Always.

Pick out any two, and he could always tell you which was more important, more valuable to him, than the other.

On the bottom were idiots. The ignorant. Anderson.

At the top, of course, was The Work. It was always The Work, ever since he became a consulting detective. Before that, even: he had Work then, too, even if he lacked his current title.

The appearance of one John Watson, however, almost made a mere semilattice of Sherlock’s mind, because suddenly there were two things—two things that settled into a top spot. Sherlock had assumed he would choose The Work first, always, but then there was the interesting case he put off because John was ill. In the end, Lestrade solved it without his help, before John recovered.

Not that John _knew_ Sherlock had turned down the case simply to remain at 221B and watch over him. No, as far as John knew, Sherlock was only home that day because he was doing an important experiment about exposure to heat on the liver Molly had given him for his birthday the week before (or what was left of it, anyway). As far as John knew, Sherlock only kept coming and going from John’s room because Sherlock was rather rudely using John’s pants for some mysterious, unspoken-of component of the experiment.

“Are you taking my temperature?” John had asked at one point, his voice thick with sleep and mucous.

“I’m testing the thermometer I’m using,” Sherlock explained. He removed John’s own thermometer from beneath John’s tongue. “I know this one you’ve been using to take your temperature is correct after testing it on myself as well earlier. If the readings on the low end of this one,” he held up the higher-temperature thermometer for theoretical use in the heat exposure experiment, “match up, I can be reasonably more confident in its accuracy. I’m using your temperature since it’s higher. This one only goes down to 40.”

Which was rubbish, but John was clearly too dazed and delirious from troubled sleep and cough medicine to notice, because he nodded and opened his mouth for Sherlock to test the other thermometer. “I hope my temperature isn’t 40,” he mumbled.

“It’s not,” Sherlock said. “But it’s closer to 40 than mine is.”

“Yeah, well…you…cold-blooded…” John muttered, and Sherlock could not suppress a smile.

John didn’t know Sherlock was checking his temperature every hour for reasons other than something nonsensical about his experiment. John didn’t know that Sherlock had muted his mobile and brought it downstairs so that John wouldn’t be distressed or kept awake by his girlfriend’s endless text messages. John didn’t know that Sherlock had really rather wanted to play _Schön Rosmarin_ on his violin in the sitting room when he had instead done a handful of pleasant little serenades by John’s window that night under the pretense of needing to keep an eye on the new neighbors’ evening habits. It wasn’t a case, but all the data Sherlock gathered while he watched over John was useful data: the hours and days John was sick were not wasted with hand-wringing over John’s health. Work was still done. John simply didn’t know it had been arranged to accommodate Sherlock’s desire to hover about and make sure John wasn’t dying.

John also didn’t know that he had almost made Sherlock’s mind a semilattice, and so he certainly didn’t know of the solution Sherlock had reached.

Because it seemed that The Work, of course, The Work, was at least of equal value to John. And it seemed that John was at least of equal value to The Work. It was a conundrum, it was stressful, it sent Sherlock’s mind scrambling for a period of time, comparing and weighing and deciding and it was impossible.

But then the solution came to him: if the ranking system in Sherlock’s mind was a lattice it was a poset, and if the ranking system in Sherlock’s mind was a poset it was antisymmetric, and if the ranking system in Sherlock’s mind was antisymmetric, and John was worth at least as much as The Work, and The Work was worth at least as much as John, then there was no other conclusion than that the two were equivalent.

And Sherlock, of course, was married to his Work.

**Author's Note:**

> ogawaryoko has been kind enough to begin translating these stories into Chinese! You may find the translations [here](http://www.mtslash.com/viewthread.php?tid=75676).
> 
> The wonderful [AlessNox](http://archiveofourown.org/users/AlessNox/pseuds/AlessNox) was generous enough to record a podfic for this fic! You may find it [here](http://www.audiofic.jinjurly.com/mathematical-proof-for-marriage).


End file.
